= Solution
For $K=k^2\lambda_{\mathrm{De}}^2\gg1$, the logarithm in <2/b> must be large and positive. In each allowed interval this happens near a denominator zero, approached from the undamped side:
$$
u_-=-v_0-O(e^{-K}),\qquad u_{\mathrm{a}}=v_1+O(e^{-K}),\qquad u_+=V+O(e^{-K}).
$$
The constants hidden by $O(e^{-K})$ have <velocity> units. Consequently,
$$
\boxed{\omega_-\sim-kv_0,\qquad \omega_{\mathrm{a}}\sim kv_1,\qquad \omega_+\sim k(v_0+v_2-v_1).}
$$
The two outer modes have <phase velocities> just outside the occupied range and therefore have no resonant particles. The middle <electron acoustic wave> lies just inside the lower edge of the <velocity plateau>, where $f_0'=0$. All three escape <Landau damping> in this collisionless model. The stated neglect of ion response remains necessary for the middle mode as well.
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